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Find the exact value without using a calculator:

[tex]\[\tan \frac{7 \pi}{12}\][/tex]

Half-Angle Formulas:
[tex]\[\sin \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}\][/tex]
[tex]\[\cos \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}\][/tex]
[tex]\[\tan \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}\][/tex]

a) [tex]\[-\frac{\sqrt{2+\sqrt{3}}}{2}\][/tex]

b) [tex]\[-\frac{\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\][/tex]

c) [tex]\[-\frac{\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\][/tex]

d) [tex]\[\frac{\sqrt{2+\sqrt{3}}}{2}\][/tex]

Sagot :

GinnyAnsweridn
To find the exact value of [tex]\(\tan \frac{7\pi}{12}\)[/tex], we can use angle sum identities.

First, note that:
[tex]\[ \frac{7\pi}{12} = \frac{3\pi}{12} + \frac{4\pi}{12} = \frac{\pi}{4} + \frac{\pi}{3} \][/tex]

We use the tangent angle sum identity:
[tex]\[ \tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \][/tex]

Set [tex]\( A = \frac{\pi}{4} \)[/tex] and [tex]\( B = \frac{\pi}{3} \)[/tex].

Now, we know:

[tex]\[ \tan\left(\frac{\pi}{4}\right) = 1 \][/tex]
[tex]\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \][/tex]

Using the tangent sum identity:
[tex]\[ \tan \left( \frac{7\pi}{12} \right) = \tan \left( \frac{\pi}{4} + \frac{\pi}{3} \right) \][/tex]

[tex]\[ \tan \left( \frac{\pi}{4} + \frac{\pi}{3} \right) = \frac{\tan \left( \frac{\pi}{4} \right) + \tan \left( \frac{\pi}{3} \right)}{1 - \tan \left( \frac{\pi}{4} \right) \tan \left( \frac{\pi}{3} \right)} \][/tex]

Substitute the known values:
[tex]\[ \tan \left( \frac{\pi}{4} + \frac{\pi}{3} \right) = \frac{1 + \sqrt{3}}{1 - 1 \cdot \sqrt{3}} \][/tex]

Solve the denominator:
[tex]\[ 1 - \sqrt{3} \][/tex]

So:
[tex]\[ \tan \left( \frac{\pi}{4} + \frac{\pi}{3} \right) = \frac{1 + \sqrt{3}}{1 - \sqrt{3}} \][/tex]

To simplify this, multiply the numerator and the denominator by the conjugate of the denominator [tex]\( 1 + \sqrt{3} \)[/tex]:
[tex]\[ \frac{(1 + \sqrt{3})(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} \][/tex]

Expand the numerator:
[tex]\[ (1 + \sqrt{3})(1 + \sqrt{3}) = 1 + 2\sqrt{3} + 3 = 4 + 2\sqrt{3} \][/tex]

Expand the denominator using the difference of squares:
[tex]\[ (1 - \sqrt{3})(1 + \sqrt{3}) = 1 - (\sqrt{3})^2 = 1 - 3 = -2 \][/tex]

So, the tangent value becomes:
[tex]\[ \frac{4 + 2\sqrt{3}}{-2} = -2 - \sqrt{3} \][/tex]

This simplified form confirms that the exact value is:
[tex]\[ \tan \left( \frac{7\pi}{12} \right) = -2 - \sqrt{3} \][/tex]

Thus, the correct option that matches this value is:
[tex]\[ \boxed{\text{c) } -\frac{\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}} \][/tex]

However, I must correct myself. The derivation using the conjugate multiplication seems to have omitted a step where the radical simplification leads to another format, making sure it conforms exactly to the given choices. The correct matching form among the choices is:

[tex]\[ \boxed{ -\frac{\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}} \][/tex]

Thus, the suitable correct option based on value conformance is indeed option (b).
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